Dissipative Abelian Sandpile Models

TL;DR

This paper introduces a family of dissipative abelian sandpile models on finite lattices, analyzes their correlation decay and avalanche behavior, and explores the critical limit as dissipation vanishes, revealing a universal critical exponent.

Contribution

The paper develops a new class of dissipative abelian sandpile models with explicit correlation decay formulas and investigates their critical behavior as dissipation approaches zero.

Findings

Correlation length decays exponentially with dissipation rate.

Critical exponent for correlation length is 1/2 in all dimensions ≥ 2.

Models connect to Potts model in the zero-field limit.

Abstract

We introduce a family of abelian sandpile models with two parameters $n, m \in {\bf N}$ defined on finite lattices on $d$-dimensional torus. Sites with $2dn+m$ or more grains of sand are unstable and topple, and in each toppling $m$ grains dissipate from the system. Because of dissipation in bulk, the models are well-defined on the shift-invariant lattices and the infinite-volume limit of systems can be taken. From the determinantal expressions, we obtain the asymptotic forms of the avalanche propagators and the height-$(0,0)$ correlations of sandpiles for large distances in the infinite-volume limit in any dimensions $d \geq 2$. We show that both of them decay exponentially with the correlation length $$ \xi(d, a)=(\sqrt{d} \sinh^{-1} \sqrt{a(a+2)} \ )^{-1}, $$ if the dissipation rate $a =m/(2dn)$ is positive. By considering a series of models with increasing $n$, we discuss the limit $a \downarrow 0$ and the critical exponent defined by $\nu_{a}=- \lim_{a \downarrow 0} \log \xi(d, a)/ \log a$ is determined as $$ \nu_{a}=1/2 $$ for all $d \geq 2$. Comparison with the $q \downarrow 0$ limit of $q$-state Potts model in external magnetic field is discussed.